#8463543

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,3)(6,7)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(26,32)(27,28)(34,36)(35,38);(1,2,5,3)(4,6,8,7)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,26)(27,33)(28,37)(29,34)(30,35)(31,32)(36,39)(38,40);(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,27,31,28)(26,33,32,37)(29,35,39,38)(30,36,40,34);(1,2,5,3)(4,7,8,6)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,28,31,27)(26,33,32,37)(29,38,39,35)(30,36,40,34);(1,3,5,2)(4,7,8,6)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,29,31,39)(26,34,32,36)(27,35,28,38)(30,37,40,33);(1,4,5,8)(2,6,3,7)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,30)(26,35)(27,36)(28,34)(29,37)(31,40)(32,38)(33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370739);